Narrative identity isn’t really an account of numerical identity, but is dependent on it. It is analogous to the PV4, but is closer to what most non-philosophers usually mean by “personal identity”; saying someone is “no longer the same person” implies such a change in personality that their life no longer fits into a single narrative.

what makes some feature mine may actually be making reference to a non-numerical type of identity, a type of identity we are thinking of when addressing the familiar question of an identity crisis: “Who am I really?” This is the question of identity as proper attributability, as providing an account of one's true self and the various attributes genuinely belonging to it.

the Narrative Criterion of Personal Identity: what makes an action, experience, or psychological characteristic properly attributable to some person (and thus a proper part of his or her true identity) is its correct incorporation into the self-told story of his or her life. … Narrative identity is thus really about a kind of psychological unity, but not just an artless or random unity.

for that subject of experiences to be a person, a genuine moral agent, those experiences must be actively unified, must be gathered together into the life of one narrative ego by virtue of a story the subject tells that weaves them together, giving them a kind of coherence and intelligibility they wouldn't otherwise have had. This is how the various experiences and events come to have any real meaning at all — rather than being merely isolated events — by being part of a larger story that relates them to one another within the context of one life

What explains my special sort of concern for myself is that I'm in fact an extended narrative ego — not some time-slice concerned about the well-being of some future time-slice — and I'm constantly extending that narrative into the future, so my concern is global, a concern for the whole self I'm creating via this story, the whole self whose various parts are mine.

what makes some past action mine (for which I'm eligible for praise or blame) is that it flowed from my central values, beliefs, and experiences, that there's a coherent story I may tell uniting it to the other elements of my life.

Shoemaker sees a problem:-

What matters to us with respect to all of our practical concerns is that we ourselves continue to exist: it's a necessary presupposition of my rational anticipation, self-concern, possibilities for compensation, and so on that I myself persist, but this is an issue of numerical identity. Another way to put this is that one can't be a person, on the narrative view, unless one gathers up the various experiences one has as a subject of experiences into a coherent narrative, but then the identity of that subject of experiences must be preserved across time for its experiences to be so gathered up.

If narrative identity depends on numerical identity, then it is as subject to fission-problems5 as accounts of numerical identity.

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